A block of wood of mass `9.8 kg` is suspended by a string. A bullet of mass `200 gm` strikes horizontally with a velocity of `100 ms^(-1)` and gets embedded in it. The maximum height attained by the block is `(g = 10 ms^(-2))`.

Options

A. 0.1 m
B. 0.2 m
C. 0.3 m
D. 0 m

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Problem
We have a block of wood with a mass of \(9.8 \, \text{kg}\) suspended by a string. A bullet of mass \(200 \, \text{g}\) (which is \(0.2 \, \text{kg}\)) strikes the block horizontally at a velocity of \(100 \, \text{m/s}\) and embeds itself in the block. We need to find the maximum height attained by the block after the collision.

### Step 2: Calculate the Initial Momentum
Before the collision, the momentum of the system is only due to the bullet since the block is at rest. The momentum \(p\) of the bullet can be calculated as:
\[ p_{\text{initial}} = m_{\text{bullet}} \cdot v_{\text{bullet}} = 0.2 \, \text{kg} \cdot 100 \, \text{m/s} = 20 \, \text{kg m/s} \]

### Step 3: Calculate the Final Momentum
After the bullet embeds itself in the block, the total mass of the system becomes:
\[ m_{\text{total}} = m_{\text{block}} + m_{\text{bullet}} = 9.8 \, \text{kg} + 0.2 \, \text{kg} = 10 \, \text{kg} \]
Let \(v\) be the velocity of the combined mass after the collision. According to the conservation of momentum:
\[ p_{\text{initial}} = p_{\text{final}} \implies 20 \, \text{kg m/s} = m_{\text{total}} \cdot v \]
Substituting the total mass:
\[ 20 = 10 \cdot v \implies v = \frac{20}{10} = 2 \, \text{m/s} \]

### Step 4: Calculate the Maximum Height
After the collision, the block and bullet move together and will rise to a maximum height where all kinetic energy is converted to potential energy. The kinetic energy (KE) just after the collision is given by:
\[ KE = \frac{1}{2} m_{\text{total}} v^2 = \frac{1}{2} \cdot 10 \cdot (2)^2 = \frac{1}{2} \cdot 10 \cdot 4 = 20 \, \text{J} \]
The potential energy (PE) at the maximum height \(h\) is given by:
\[ PE = m_{\text{total}} \cdot g \cdot h \]
Setting the kinetic energy equal to the potential energy at maximum height:
\[ 20 = 10 \cdot 10 \cdot h \]
Solving for \(h\):
\[ 20 = 100h \implies h = \frac{20}{100} = 0.2 \, \text{m} \]

### Final Answer
The maximum height attained by the block is \(0.2 \, \text{m}\).